Layer | Fill | Outline |
---|
Map layers
Theme | Visible | Selectable | Appearance | Zoom Range (now: 0) |
---|
Fill | Stroke |
---|---|
Collaborating Authors
Abstract Many scholars have used the boundary method to solve pressure transient problems of geometrically complex oil reservoirs. Their research mainly concentrates in oil reservoirs. However, the research on the pressure transient problems of geometrically complex gas reservoirs is very few. The description of pressure transient problems of gas reservoirs is more complex than that of oil reservoirs as the compressibility of gas and the variability of gas viscosity. By deriving rigorously in detail, this paper develops the Laplace space boundary elementary method for solving pressure transient problems in geometrically complex gas reservoirs. The corresponding boundary element models are built and solved. Type curves are computed and drawn. The characteristics of the typical curves are analyzed on the basis of fluid flow mechanism and flow process. The results of this paper not only help us to understand the fluid flow mechanism in this kind of geometrically complex gas reservoir but also can be applied to well testing. The results provide theoretical basis for people to scientifically and economically exploiting reservoirs. Besides, this paper theoretically proves the versatility of BEM in oil reservoir and gas reservoir seepage problems. Introduction Compared with analytical method, boundary element method (BEM) can solve more complicated flow problems. Such as in these geometrically complex reservoir: irregularly shaped reservoir, composite reservoir with some wells located in inner region and some wells located in outer region, reservoir with some section of impermeable boundary and some section of constant pressure boundary, reservoir with multiple-well problem in irregularly shaped reservoir, and so on. The foundation of Boundary element method is Green Function theory. In 1973, Gringarten and Ramey introduced the Green Function method into research of unsteady flow problem, and this paved the way for the application of boundary element method to research of reservoir fluid flow problem. In 1977, Liggettd and Liu firstly used the boundary element method to study fluid flow problem through porous media. In 1979, Liggettd and Liu compared the solution of laplace space with solution of real space. In 1986, Numbere and Tiab used BEM as a useful streamline generating technique for steady state water injection problems. Some time-dependent problems governed by diffusivity equation have also been researched by use of the BEM. In 1989, Kikani and Horne discussed the pressure response when a well or several wells produce in impermeable square or triangular oil reservoirs. They analyzed the characteristics of the semi-log pressure curves and the log-log pressure derivative curves. By comparing the BEM solution with the analytical solution, they obtained good results. In 1992, Kikani and Horne analyzed the pressure response of arbitrarily shaped oil reservoir with the BEM. In 1992, Zhang and Zeng used the Boundary element Method to analyze Pressure-transient response of irregular shaped double-porosity reservoirs. In heterogeneous media, there is no fundamental solution in a closed form. To overcome this difficulty, Sato and Horne composed the governing equation into various order perturbation equations, for which the free-space Green's function can be found. Then at each level of perturbation, the solution is computed by the BEM and the summation of various order perturbation solutions gives the complete solution for the original governing equation. Jongkittinarukom and Tiab develop the BEM for analyzing pressure transient problems of a horizontal well in multilayer-resevoir. Almost all of these studies focus on the pressure transient problems of oil reservoirs, and the studied fluid is either incompressible or slightly compressible. However, the studies to the compressible fluid are difficult to find. By deriving rigorously in detail, the paper develops the Laplace space BEM for solving pressure transient problems in geometrically complex gas reservoirs.
- Asia > China (0.28)
- North America > Canada (0.28)
- South America > Colombia > T Formation (0.99)
- Asia > Middle East > Turkey > Selmo Field > S-2 Well (0.99)
Abstract Analytical solutions in the petroleum engineering literature have always been treated as benchmarks for many complicated situations in reservoirs. But efforts to develop analytical solutions for some transient-flow problems have been hindered seriously due to a lack of appropriate mathematical tools. Nevertheless, no systematic attempts to take advantage of using an integral-transform technique (ITT) have been reported in the literature either. This study demonstrates that the use of the ITT results in the development of a huge number of useful solutions to transient-flow problems in a Cartesian coordinate system. New analytical solutions are presented comprehensively in a compact, convenient form. For instance, the closedform solutions in terms of the time and space variables can be written down for all 729 possible combinations of the conditions at the boundaries with constant-potential, no-flow and mixed boundary conditions in a parallelepiped. The parameters pertinent to the solutions are presented in tables. The generality of the solutions is greatly enhanced by considering the production rates through the wells to be time dependent. This situation is treated in a general but computationally efficient way. Thus, the use of the solutions is permissible even if there has been a production history in a system. Example problems with solutions are presented to illustrate the methodology. The new solutions are useful for analyzing drawdown-, buildup- and interference-test data. Introduction Mikhailov and Ozisik have suggested that analytical solutions, when available, are advantageous in that they provide good insights into the significance of various parameters in the system affecting the transport phenomena, as well as accurate benchmarks for a numerical approach. Moreover, analytical solutions in the petroleum engineering literature have always been treated as benchmarks for many complicated situations in reservoirs. Also, these solutions have been used to complement numerical approaches to solving some of these problems. But efforts to develop analytical solutions to most transient-flow problems have been hindered due to a lack of easy-to-deal-with and appropriate mathematical tools. For example, the closed-form solution for flow due to production through a partially penetrating well from a finite, cylindrical-radial reservoir has not been presented in the literature. Finally, in the well-testing literature, analytical solutions are used to identify different flow regimes for computing different rock and fluid properties. Sporadic attempts to use the Fourier transforms for finite domains to develop solutions for some transientflow problems are reported in the literature. In the reported studies, one-dimensional (1D) and twodimensional (2D), readily-solvable problems in finite domains were dealt with. Rahman used the Fourier sine and cosine transforms to develop transient solutions for a rectangular parallelepiped for two sets of boundary conditions. Another popular method, the Laplacetransform technique, can be used only in semi-infinite domains; thus, it is limited to the time and semi-infinite space domains. Rahman and Ambastha successfully showed that the analytical solutions for transient flow in compartmentalized reservoirs can be developed using the ITT for finite, composite domains. Despite this development, to the best of the authors' knowledge, no systematic attempts have been made to exploit the advantages of using the ITT for finite, homogeneous domains.
Summary Two boundary-element-method (BEM) formulations are proposed for the solutionof pressure-transient problems in proposed for the solution ofpressure-transient problems in homogeneous, anisotropic reservoirs. Pressuresolutions in arbitrary reservoir shapes with multiple sources and/or sinks anda variety of constant and/or time-dependent boundary conditions can begenerated. This technique is superior to numerical methods because it preservesthe analytical nature of the solution and because numerical dispersion andgrid-orientation effects are nonexistent. Procedures for the convolution and Laplace-domain solution procedures are compared, and problems illustrating thevarious aspects of the BEM are solved. Introduction A considerable amount of time and effort is spent analyzing and interpretingpressure data collected from wells to determine reservoir parameters so thatreservoir performance can be predicted. Pressure data are matched to an assumedmodel whose behavior Pressure data are matched to an assumed model whosebehavior is known by analytical or numerical means. Once a reasonable match isobtained, the parameters are evaluated against the solution of the assumedmodel. The basic requirement is that reasonably accurate solutions can begenerated by analytical or numerical means for the particular reservoir inquestion. Analytical methods have several deficiencies. One of the major shortcomingsis that only a few simple geometries lend themselves to the limited number ofexisting solution procedures. Other simple boundary shapes can be generated bysuperposing image wells of a particular type. These methods have been studiedextensively. Earlougher et al. and Earlougher discuss the solution andinterpretation of pressure behavior in bounded rectangular systems. Larsengives a general algorithm for the location of images to generate solutions in afew regular boundary shapes. Extension of this methodology, however, does notnecessarily carry over to other regular and irregular shapes. The image method of generating solutions to various types of boundaryconditions may become too tedious for calculations at long times when theimages fill the entire space and consequently require considerable computingeffort. Also, mixed-boundary-value problems (i.e., problems in which differenttypes of boundary problems (i.e., problems in which different types of boundaryconditions are specified on one section of the boundary) cannot be treated bythese techniques. An example of a mixed-boundary-value problem is the symmetryelement of a constant-pressure, vertical problem is the symmetry element of aconstant-pressure, vertical fracture in a closed square. In such numerical methods as finite differences and finite elements, thegoverning equations are solved approximately in the solution domain. Finite-difference methods, which are widely used in the oil industry, haveproblems with numerical dispersion, grid-orientation effects, and conformanceto the boundaries. In recent years, the boundary integral equation method, alsoknown as BEM, has been considered as a viable alternative to finite-differenceand finite-element methods for the solution of linear differential equationsover irregular boundaries. The advantage of the technique stems from thereduction in the dimensionality of the problem by one; i.e., a 3D problem isreduced to a 2D problem and so on (this advantage is not always realized ifdistributed sources or nonhomogeneous initial conditions are present in thedifferential equation). Also, because it is a present in the differentialequation). Also, because it is a surface or boundary procedure (as a result ofthe reduction in dimensionality), the BEM conforms well to the boundaries. In the BEM, the governing linear differential operator is solved exactlywithin an arbitrarily shaped problem domain. Approximations are made only onthe boundaries. The exact solution of the governing operator in terms of theapproximation at the boundaries provides an analytical flavor to the method andretains the accuracy of the solutions. Thus, the grid-orientation effects andnumerical dispersion suffered by the finite-difference and finite-elementmethods are eliminated. Also, because it is a smoothing procedure, theresulting integrals, even if they cannot be performed analytically, incurminimal error when accurate Gaussian-type, panel-integration schemes areused. The early development of this technique took place within the realm ofpotential theory. The horizons of the method expanded tremendously during the1970's in such engineering disciplines as elastodynamics, heat transfer, fluidmechanics, and groundwater hydrology. Since then, many studies have beenconducted to solve various physical problems. In petroleum engineeringapplications, Numbere and Tiab and Masukawa and Horne used the BEM as astreamline-generating technique for steady-state, water-injection problems forfavorable mobility ratios. Different methods of solving time-dependent problemsgoverned by the diffusivity equation have also been considered, with eachmethod having some useful aspects and drawbacks. Before discussing the specifics of any particular formulation, we outlinethe basic development of a BEM procedure. The governing linear differentialoperator is cast into an integral equation form by the use of the fundamentalsolution of an adjoint operator, which is also known as the free-space Green'sfunction or the derivative function. With Green's identity, the integrals arecast in terms of boundary and domain integrals. The domain integrals representthe inhomogeneity in the governing equation in terms of initial condition orsource terms. The boundary of the problem is then discretized in elements, onwhich interpolation functions are defined. The integral equation is used firstto evaluate the unknown boundary data and then used as quadratures to evaluatesolutions at any interior point. Two formulations, the convolution BEM and the Laplace-space BEM arepresented next. The differences between the two methods and between them andother published formulations are highlighted, and the verification of bothformulations with simple problems that have analytical solutions are shown. Afew practical pressure-transient problems are solved to show the efficacy ofthe pressure-transient problems are solved to show the efficacy of thetechniques. Problem Definition Problem Definition Fig. 1 shows a typical reservoirgeometry. The domain of the problem is defined by, and the correspondingbounding surface is problem is defined by, and the corresponding boundingsurface is represented by on sections of which different types of boundaryconditions could be applied. The nondimensionalized equation for 2D flow of aslightly compressible, single-phase fluid in a homogeneous, anisotropic, finitereservoir in the Cartesian coordinate system, with the coordinate axes alignedwith the principal permeability directions, is permeability directions, is ..........................(1) SPEFE P. 53
Green's function and source functions are used to solve 2D and 3D transient flow problems that may result from complex well geometries, such as partially penetrating vertical and inclined wells, hydraulically fractured wells, and horizontal wells. The point-source solution was first introduced by Lord Kelvin[1] for the solution of heat conduction problems and was extensively discussed by Carslaw and Jaeger.[2] The point-source solution is usually obtained by finding the limiting form of the pressure drop resulting from a spherical source as the source volume vanishes. In our terminology, a source is a point, line, surface, or volume at which fluids are withdrawn from the reservoir. Strictly speaking, fluid withdrawal should be associated with a sink, and the injection of fluids should be related to a source. Here, however, the term source is used for both production and injection with the convention that a negative withdrawal rate indicates injection.
- North America > United States (0.47)
- Europe > United Kingdom > England (0.47)
- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)
Abstract Fluids are injected in subsurface permeable formations for various purposes including waste disposal, gas storage, CO2 sequestration, and enhanced oil/gas recovery. Containment of the injected fluids is needed to meet the regulatory requirements and/or to ensure efficiency of the intended processes. The injected fluids can leak to overlying formations in presence of leakage pathways. Improperly plugged and abandoned (P&A) wells are considered as the main potential leakage pathways. In a previous work, we introduced a vertical pressure transient interference test and presented an analysis methodology to detect and characterize leaking wells. The analysis methodology was based on an inverse modeling algorithm that can be highly instable and computationally expensive. Here, we propose an easy-to-use fully graphical methodology to characterize leaking wells. The pressure measurements are graphed in three different forms. The slopes and intercepts of the line-fitted graphs are used to determine the leak location and transmissibility as well as the transmissivity ratio of the connected zones. The graphical method is applied to an example problem to illustrate its application procedure and effectiveness. Introduction The Leakage through abandoned wells and improperly plugged boreholes can create vertical communication between otherwise hydrologically isolated permeable zones. The driving mechanism behind the leakage can be the hydraulic gradients created by injection into one of the zones. Zeidouni and Pooladi-Darvish (2012a, 2012b) introduced a vertical interference test to detect and characterize a leaking well connecting the operating zone to an overlying non-operating zone which is otherwise separated by a sealing caprock. The test involves injection (production) into (from) the operating zone (OZ) and observing the pressure at a distance both in the OZ and the monitoring zone (MZ). We use injection throughout this paper for consistency. Several researchers attempted to analyze the pressure observations through inverse modelling approach and data assimilation (Wang and Small 2014, Jung, Zhou, and Birkholzer 2013, Sun et al. 2013, Zeidouni and Pooladi-Darvish 2012a, b, Chabora and Benson 2009, Jung, Zhou, and Birkholzer 2015, Keating et al. 2014). While inverse modeling can be very useful, it requires robust and computationally expensive inversion techniques that may not be easy to implement in practice. Also, inverse models can be very instable if the unknown parameters are not fully independent. It would be useful to develop graphical approaches such as those used in conventional pressure transient analysis that can be conveniently used in analyzing the pressured data for leaking well characterization.